In $\mathbb C^n$, we can define the inner product between $u=\{u_1,\ldots,u_n\}$ and $v=\{v_1,\ldots,v_n\}$ as $\langle u,v\rangle=u_1\overline{v_1}+\ldots+u_n\overline v_n$. I've read in a book that we can define in $\mathbb C^n$ the inner product $\langle u,v\rangle=u^*v$, where $u^*$ is the Hermitian matrix of $u$, the problem is these inner products don't match, my question is these inner products are different from each other?
Thanks in advance
Really $u^*$ is nothing more than the conjugate transpose, i.e. if $u=\left(\begin{array}{c} u_1 \\ \vdots \\ u_n\end{array}\right)$, then $u^* = (\begin{array}{c} \overline{u_1} & \cdots & \overline{u_n}\end{array})$. So at that point $u^*v$ is just the regular dot product (which gives $\langle u,v\rangle = u^*v = \overline{u_1}v_1+\cdots+\overline{u_n}v_n$) and so the resulting inner product is nearly identical to the first one in your post except that the complex conjugate is opposite. For all practical purposes these inner products are identical (results you get from the first correspond to equivalent results from the other). So while the expressions are different, they're really just two sides of the same coin if you want to think of it like that.
The choice of where to put the conjugate has historically been the focus of a bit of debate between physicists and mathematicians. Physicists most often conjugate the first vector whereas mathematicians like to conjugate the second vector but it is a matter of style rather than substance as there is no reason to choose one over the other. All it amounts to is saying that inner products are linear in the second argument and conjugate linear in the second argument versus inner products are linear in the first argument and conjugate linear in the second argument (resp. from physicists and mathematicians). Personally, I choose to put the conjugate on $v$ rather than $u$ but to each their own.
